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Mathematics of Computation

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Evaluating Jacquet’s $\mathbf {\textrm {GL}(n)}$ Whittaker function
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by Kevin A. Broughan PDF
Math. Comp. 78 (2009), 1061-1072 Request permission

Abstract:

Algorithms for the explicit symbolic and numeric evaluation of Jacquet’s Whittaker function for the $GL(n,\mathbb {R})$ based generalized upper half-plane for $n\ge 2$, and an implementation for symbolic evaluation in the Mathematica package GL(n)pack, are described. This requires a comparison of the different definitions of Whittaker function which have appeared in the literature.
References
  • Peter A. Becker, On the integration of products of Whittaker functions with respect to the second index, J. Math. Phys. 45 (2004), no. 2, 761–773. MR 2029096, DOI 10.1063/1.1634351
  • Dorian Goldfeld, Automorphic forms and $L$-functions for the group $\textrm {GL}(n,\mathbf R)$, Cambridge Studies in Advanced Mathematics, vol. 99, Cambridge University Press, Cambridge, 2006. With an appendix by Kevin A. Broughan. MR 2254662, DOI 10.1017/CBO9780511542923
  • Solomon Friedberg, Poincaré series for $\textrm {GL}(n)$: Fourier expansion, Kloosterman sums, and algebreo-geometric estimates, Math. Z. 196 (1987), no. 2, 165–188. MR 910824, DOI 10.1007/BF01163653
  • Dorian Goldfeld, Automorphic forms and $L$-functions for the group $\textrm {GL}(n,\mathbf R)$, Cambridge Studies in Advanced Mathematics, vol. 99, Cambridge University Press, Cambridge, 2006. With an appendix by Kevin A. Broughan. MR 2254662, DOI 10.1017/CBO9780511542923
  • I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 6th ed., Academic Press, Inc., San Diego, CA, 2000. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. MR 1773820
  • Taku Ishii, A remark on Whittaker functions on $\textrm {SL}(n,\Bbb R)$, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 2, 483–492. MR 2147897
  • Hervé Jacquet, Fonctions de Whittaker associées aux groupes de Chevalley, Bull. Soc. Math. France 95 (1967), 243–309 (French). MR 271275
  • N. N. Lebedev, Special functions and their applications, Dover Publications, Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman; Unabridged and corrected republication. MR 0350075
  • Frank W. J. Olver, Asymptotics and special functions, AKP Classics, A K Peters, Ltd., Wellesley, MA, 1997. Reprint of the 1974 original [Academic Press, New York; MR0435697 (55 #8655)]. MR 1429619
  • I. I. Pjateckij-Šapiro, Euler subgroups, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 597–620. MR 0406935
  • J. A. Shalika, The multiplicity one theorem for $\textrm {GL}_{n}$, Ann. of Math. (2) 100 (1974), 171–193. MR 348047, DOI 10.2307/1971071
  • I. H. Sloan and T. R. Osborn, Multiple integration over bounded and unbounded regions, J. Comput. Appl. Math. 17 (1987), no. 1-2, 181–196. MR 884269, DOI 10.1016/0377-0427(87)90046-X
  • Eric Stade, On explicit integral formulas for $\textrm {GL}(n,\textbf {R})$-Whittaker functions, Duke Math. J. 60 (1990), no. 2, 313–362. With an appendix by Daniel Bump, Solomon Friedberg and Jeffrey Hoffstein. MR 1047756, DOI 10.1215/S0012-7094-90-06013-2
  • Eric Stade, Mellin transforms of $\textrm {GL}(n,\Bbb R)$ Whittaker functions, Amer. J. Math. 123 (2001), no. 1, 121–161. MR 1827280
  • Eric Stade, Archimedean $L$-factors on $\textrm {GL}(n)\times \textrm {GL}(n)$ and generalized Barnes integrals, Israel J. Math. 127 (2002), 201–219. MR 1900699, DOI 10.1007/BF02784531
  • A. H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0327006
  • Audrey Terras, Harmonic analysis on symmetric spaces and applications. I, Springer-Verlag, New York, 1985. MR 791406, DOI 10.1007/978-1-4612-5128-6
  • E. T. Whittaker and G. N. Watson, A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions, Cambridge University Press, New York, 1962. Fourth edition. Reprinted. MR 0178117
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Additional Information
  • Kevin A. Broughan
  • Affiliation: Department of Mathematics, University of Waikato, Hamilton, New Zealand
  • Email: kab@waikato.ac.nz
  • Received by editor(s): November 6, 2006
  • Received by editor(s) in revised form: March 3, 2008
  • Published electronically: August 28, 2008
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 78 (2009), 1061-1072
  • MSC (2000): Primary 33C15, 22E30, 11E57, 11E76
  • DOI: https://doi.org/10.1090/S0025-5718-08-02158-3
  • MathSciNet review: 2476570